Inequality 3 | 2x-2 | +16<58
•simplify as much as possible without moving the absolute value sign
•rewrite the inequality as two inequalities with out absolute value signs• find solution set of the inequality and graph solution set on a number line
subtract 16 from each side, then divide each side by 3
abs(2x-2)<14
what if 2x-2<14, then x<8
what if -(2x-2)<14, then
-2x+4<14
-2x<10
x>-5
0.08x+2.1=
To simplify the inequality without moving the absolute value sign, we can start by removing the absolute value around the expression 2x - 2:
|2x - 2| = 2x - 2 (since 2x - 2 is non-negative)
Now, let's substitute this into the inequality:
3(2x - 2) + 16 < 58
Simplifying further:
6x - 6 + 16 < 58
Combine like terms:
6x + 10 < 58
To rewrite the inequality as two inequalities without absolute value signs, we need to consider two cases:
Case 1: When 2x - 2 is positive (greater than or equal to zero), the absolute value does nothing, so we can rewrite the inequality as:
3(2x - 2) + 16 < 58
6x - 6 + 16 < 58
6x + 10 < 58
Case 2: When 2x - 2 is negative (less than zero), the absolute value changes the sign, so we need to negate the expression inside the absolute value:
3(-2x + 2) + 16 < 58
-6x + 6 + 16 < 58
-6x + 22 < 58
Now, we have two inequalities:
6x + 10 < 58 (for the positive case)
-6x + 22 < 58 (for the negative case)
To find the solution set, solve each inequality separately:
For 6x + 10 < 58:
Subtract 10 from both sides:
6x < 48
Divide both sides by 6 (since the coefficient of x is positive):
x < 8
For -6x + 22 < 58:
Subtract 22 from both sides:
-6x < 36
Divide both sides by -6 (since the coefficient of x is negative, we reverse the inequality sign):
x > -6
Taking the intersection of the two solutions, we find the overall solution set:
-6 < x < 8
To graph the solution set on a number line, draw a line and mark -6 and 8 as open circles. Then shade the area between these two points, indicating that x lies within this range.